Orthogonal time frequency space communication system compatible with OFDM

1. A method of transmitting data over a communication channel, the method comprising:

receiving a plurality of information symbols;

encoding an N×M array containing the plurality of information symbols into a two-dimensional array of modulation symbols by spreading each of the plurality of information symbols using a set of two-dimensional basis functions that are mutually orthogonal with respect to both time and frequency; and

transmitting the two-dimensional array of modulation symbols using M mutually orthogonal waveforms included within M frequency sub-bands.

2. A method of transmitting data over a communication channel, the method comprising:

receiving a plurality of information symbols;

encoding an N×M array containing the plurality of information symbols into a two-dimensional array of modulation symbols by spreading each of the plurality of information symbols with respect to both time and frequency; and

transmitting the two-dimensional array of modulation symbols using M mutually orthogonal waveforms included within M frequency sub-bands,

wherein the encoding includes:

transforming the N×M array into an array of filtered orthogonal frequency division multiplexing (OFDM) symbols using at least one Fourier transform and a filtering process; and

transforming the array of filtered OFDM symbols into an array of orthogonal time frequency space (OTFS) symbols using at least one two-dimensional Fourier transform wherein the array of OTFS symbols corresponds to the two-dimensional array of modulation symbols.

3. The method of claim 2 , wherein the transmitting includes using a set of M narrow-band filters to produce the M waveforms.

4. The method of claim 1 , wherein the encoding includes combining an inverse symplectic transform with a windowing operation.

5. The method of claim 1 , wherein the encoding is performed in accordance with the following relationship:

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wherein x[l,k], k=0, . . . , N−1, l=0, . . . , M−1 represents the N×M array containing the plurality of information symbols, X[n,m], n=0, . . . , N−1, m=0, . . . , M−1 represents the two-dimensional array of modulation symbols, Wt,[n,m] is a windowing function, and b k,l [m, n] represents a set of basis functions.

6. An apparatus for wireless communication over a data channel, comprising:

a processor configured to:

receive a plurality of data symbols;

encode an N×M two-dimensional array containing the plurality of data symbols into a two-dimensional array of modulation symbols by spreading each of the plurality of data symbols using a set of cyclically time-shifted and frequency-shifted basis functions; and

transmit the two-dimensional array of modulation symbols using M mutually orthogonal wireless waveforms included within M frequency sub-bands.

7. The apparatus of claim 6 , wherein the processor is configured, as part of encoding the N×M two-dimensional array, to:

transform the N×M two-dimensional array of data symbols into an array of filtered OFDM symbols using at least one Fourier transform and a filtering process; and

transform the array of filtered OFDM symbols into an array of OTFS symbols using at least one two-dimensional Fourier transform wherein the array of OTFS symbols corresponds to the two-dimensional array of modulation symbols.

8. The apparatus of claim 6 , wherein the processor is configured, as part of encoding the N×M two-dimensional array, to:

encode the N×M two-dimensional array containing the plurality of data symbols onto at least one symplectic-like analysis compatible manifold distributed over a column time axis of length T and row frequency axis of length F, thereby producing at least one information manifold; and

transform the at least one information manifold in accordance with a two-dimensional symplectic-like Fourier transform, thereby producing at least one two-dimensional Fourier transformed information manifold.

9. The apparatus of claim 8 , wherein the transmitting includes transmitting each at least one two-dimensional Fourier transformed information manifold by selecting, over all frequencies and times of said two-dimensional Fourier transformed information manifold, a transmitting time slice of duration proportional to Tμ, where μ=1/N, and passing those frequencies in said two-dimensional Fourier transformed information manifold corresponding to said transmitting time slice through a bank of at least M different, non-overlapping, narrow-band frequency filters, and transmitting resulting filtered waveforms as a plurality of at least M simultaneously transmitted mutually orthogonal waveforms, over different transmitted time intervals, until an entire two-dimensional Fourier transformed information manifold has been transmitted.

10. The apparatus of claim 6 , wherein the processor is further configured to:

receive the M mutually orthogonal wireless waveforms;

determine a two-dimensional channel state; and

use an inverse of the encoding and the two-dimensional channel state to extract the plurality of data symbols from the M mutually orthogonal wireless waveforms.

11. The apparatus of claim 9 , wherein the processor is further configured to:

receive each said at least one two-dimensional Fourier transformed information manifold by using, over at least all frequencies and times of said two-dimensional Fourier transformed information manifold, at least one receiver processor to select a receiving time slice that is less than or equal to the duration of the transmitted time intervals, and receiving these channel convoluted waveforms on each said receiving time slice through a receiving bank of at least M different, non-overlapping, narrow-band frequency filters, and receiving said channel-convoluted waveforms over every receiving time slice until an approximation of said two-dimensional Fourier transformed information manifold has been received; and

perform at least one of:

a) using an inverse of said two-dimensional symplectic-like Fourier transform to transform said approximation of said two-dimensional Fourier transformed information manifold into an approximation of said at least one received information manifold, and using information pertaining to said two-dimensional channel state to correct said at least one received information manifold for said data channel impairments; or

b) using information pertaining to said two-dimensional channel state to correct said approximation of said two-dimensional Fourier transformed information manifold for said data channel impairments, and using the inverse of said two-dimensional symplectic-like Fourier transform to in turn produce said at least one received information manifold.

12. An apparatus for wirelessly communicating data over a data channel, comprising:

a processor configured to:

encode an N×M two-dimensional array containing a plurality of data symbols onto at least one symplectic analysis compatible manifold distributed over a column time axis of length T and row frequency axis of length F, thereby producing at least one information manifold;

transform said at least one information manifold according to a two-dimensional symplectic-like Fourier transform, thereby producing at least one two-dimensional Fourier transformed information manifold; and

transmit each at least one two-dimensional Fourier transformed information manifold by selecting, over all frequencies and times of said two-dimensional Fourier transformed information manifold, a transmitting time slice of duration proportional to Tμ, where μ=1/N, and passing those frequencies in said two-dimensional Fourier transformed information manifold corresponding to said transmitting time slice through a bank of at least M different, non-overlapping, narrow-band frequency filters, and transmitting resulting filtered waveforms as a plurality of at least M simultaneously transmitted mutually orthogonal waveforms, over different transmitted time intervals, until an entire two-dimensional Fourier transformed information manifold has been transmitted.

13. The apparatus of claim 12 , wherein the data channel includes an impairment impairing said waveforms, said impairment being at least partially describable according to a two-dimensional channel state of said channel, thereby producing channel-convoluted waveforms, and wherein the processor is further configured to:

receive each said at least one two-dimensional Fourier transformed information manifold by using, over at least all frequencies and times of said two-dimensional Fourier transformed information manifold, at least one receiver processor to select a receiving time slice that is less than or equal to the duration of the transmitted time intervals, and receiving these channel convoluted waveforms on each said receiving time slice through a receiving bank of at least M different, non-overlapping, narrow-band frequency filters, and receiving said channel-convoluted waveforms over every receiving time slice until an approximation of said two-dimensional Fourier transformed information manifold has been received; and

perform at least one of:

a) using an inverse of said two-dimensional symplectic-like Fourier transform to transform said approximation of said two-dimensional Fourier transformed information manifold into an approximation of said at least one received information manifold, and using information pertaining to said two-dimensional channel state to correct said at least one received information manifold for said data channel impairments; or

b) using information pertaining to said two-dimensional channel state to correct said approximation of said two-dimensional Fourier transformed information manifold for said data channel impairments, and using the inverse of said two-dimensional symplectic-like Fourier transform to in turn produce said at least one received information manifold; and

using said at least one receiver processor to decode and extract said data symbols from said at least one received information manifold.

14. The apparatus of claim 13 , wherein the processor is further configured to:

transform said at least one N×M two-dimensional array of data according to a spreading operation either prior or after encoding said at least one N×M two-dimensional array of data symbols onto at least one symplectic analysis compatible manifold, wherein said receiving processor uses an inverse of said spreading operation to decode and extract said data symbols from said at least one received information manifold.

15. The apparatus of claim 14 , wherein said spreading operation is convolution with a two-dimensional chirp operation.

16. The apparatus of claim 13 , wherein said non-overlapping, narrow-band frequency filters pass frequencies from said at least one two-dimensional Fourier transformed information manifold proportional to g(t·e jkω 0 ), where j is the square root of −1, t corresponds to a given time slice of duration Tμ chosen from said at least one two-dimensional Fourier transformed information manifold, and k corresponds to a given row position on said at least one two-dimensional Fourier transformed information manifold, where k varies between 1 and M.

17. The apparatus of claim 16 , wherein ω 0 is proportional to 1/T, and T=M/(allowed wireless bandwidth).

18. The apparatus of claim 12 , wherein said at least one two-dimensional Fourier transformed information manifold has overall dimensions NT μ according to a time axis and M/T according to a frequency axis, and each cell in said at least one two-dimensional Fourier transformed information manifold has overall dimensions proportional to Tμ according to a time axis and 1/T according to a frequency axis.

19. The apparatus of claim 13 , wherein the processor is further configured, as part of receiving each said at least one two-dimensional Fourier transformed information manifold, to:

receive said at least one two-dimensional Fourier transformed information manifold according to receiving time slices and receiving banks of different, non-overlapping, narrow-band frequency filters that oversamples the corresponding transmitting time slices and transmitting banks of different, non-overlapping, narrow-band frequency filters.

20. The apparatus of claim 12 , wherein the processor is further configured, as part of transmitting the two-dimensional Fourier transformed information manifold, to transmit either:

1) a plurality of at least M simultaneously transmitted mutually orthogonal waveforms over either different consecutive transmitted time intervals; or

2) a plurality OTFS data or OTFS pilot bursts comprising at least M simultaneously transmitted mutually orthogonal waveform bursts over different transmitted intervals separated by at least one spacer time interval.